3.28.11 \(\int \frac {x^2 (-2+x^8) \sqrt [4]{2-2 x^4+x^8}}{(2+x^8) (4-x^4+2 x^8)} \, dx\)

Optimal. Leaf size=248 \[ \frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^8-2 x^4+2}}{\sqrt {2} x^2-\sqrt {x^8-2 x^4+2}}\right )}{2 \sqrt [4]{2}}+\frac {\sqrt [4]{3} \tan ^{-1}\left (\frac {6^{3/4} x \sqrt [4]{x^8-2 x^4+2}}{\sqrt {6} \sqrt {x^8-2 x^4+2}-3 x^2}\right )}{2\ 2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{x^8-2 x^4+2}}{2 x^2+\sqrt {2} \sqrt {x^8-2 x^4+2}}\right )}{2 \sqrt [4]{2}}-\frac {\sqrt [4]{3} \tanh ^{-1}\left (\frac {6^{3/4} x \sqrt [4]{x^8-2 x^4+2}}{3 x^2+\sqrt {6} \sqrt {x^8-2 x^4+2}}\right )}{2\ 2^{3/4}} \]

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Rubi [F]  time = 8.72, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^2 \left (-2+x^8\right ) \sqrt [4]{2-2 x^4+x^8}}{\left (2+x^8\right ) \left (4-x^4+2 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^2*(-2 + x^8)*(2 - 2*x^4 + x^8)^(1/4))/((2 + x^8)*(4 - x^4 + 2*x^8)),x]

[Out]

-1/4*((-1)^(7/8)*Defer[Int][(2 - 2*x^4 + x^8)^(1/4)/((-2)^(1/8) - x), x])/2^(1/8) + ((-1)^(3/8)*Defer[Int][(2
- 2*x^4 + x^8)^(1/4)/(-((-1)^(5/8)*2^(1/8)) - x), x])/(4*2^(1/8)) - ((-1)^(7/8)*Defer[Int][(2 - 2*x^4 + x^8)^(
1/4)/((-2)^(1/8) + x), x])/(4*2^(1/8)) + ((-1)^(3/8)*Defer[Int][(2 - 2*x^4 + x^8)^(1/4)/(-((-1)^(5/8)*2^(1/8))
+ x), x])/(4*2^(1/8)) + Defer[Int][(2 - 2*x^4 + x^8)^(1/4)/(Sqrt[-1 + I] - 2^(1/8)*x), x]/(2*Sqrt[-1 + I]*2^(
3/4)) + Defer[Int][(2 - 2*x^4 + x^8)^(1/4)/(Sqrt[1 - I] - 2^(1/8)*x), x]/(2*Sqrt[1 - I]*2^(3/4)) + Defer[Int][
(2 - 2*x^4 + x^8)^(1/4)/(Sqrt[-1 + I] + 2^(1/8)*x), x]/(2*Sqrt[-1 + I]*2^(3/4)) + Defer[Int][(2 - 2*x^4 + x^8)
^(1/4)/(Sqrt[1 - I] + 2^(1/8)*x), x]/(2*Sqrt[1 - I]*2^(3/4)) + ((I/2)*Defer[Int][(2 - 2*x^4 + x^8)^(1/4)/(Sqrt
[-Sqrt[1 - I*Sqrt[31]]] - Sqrt[2]*x), x])/Sqrt[-31*Sqrt[1 - I*Sqrt[31]]] - ((I + Sqrt[31])*Defer[Int][(2 - 2*x
^4 + x^8)^(1/4)/(Sqrt[-Sqrt[1 - I*Sqrt[31]]] - Sqrt[2]*x), x])/(2*Sqrt[-31*Sqrt[1 - I*Sqrt[31]]]) + ((I/2)*Def
er[Int][(2 - 2*x^4 + x^8)^(1/4)/((1 - I*Sqrt[31])^(1/4) - Sqrt[2]*x), x])/(Sqrt[31]*(1 - I*Sqrt[31])^(1/4)) -
((I/2)*(1 - I*Sqrt[31])^(3/4)*Defer[Int][(2 - 2*x^4 + x^8)^(1/4)/((1 - I*Sqrt[31])^(1/4) - Sqrt[2]*x), x])/Sqr
t[31] - ((I/2)*Defer[Int][(2 - 2*x^4 + x^8)^(1/4)/(Sqrt[-Sqrt[1 + I*Sqrt[31]]] - Sqrt[2]*x), x])/Sqrt[-31*Sqrt
[1 + I*Sqrt[31]]] + ((I - Sqrt[31])*Defer[Int][(2 - 2*x^4 + x^8)^(1/4)/(Sqrt[-Sqrt[1 + I*Sqrt[31]]] - Sqrt[2]*
x), x])/(2*Sqrt[-31*Sqrt[1 + I*Sqrt[31]]]) - ((I/2)*Defer[Int][(2 - 2*x^4 + x^8)^(1/4)/((1 + I*Sqrt[31])^(1/4)
- Sqrt[2]*x), x])/(Sqrt[31]*(1 + I*Sqrt[31])^(1/4)) + ((I/2)*(1 + I*Sqrt[31])^(3/4)*Defer[Int][(2 - 2*x^4 + x
^8)^(1/4)/((1 + I*Sqrt[31])^(1/4) - Sqrt[2]*x), x])/Sqrt[31] + ((I/2)*Defer[Int][(2 - 2*x^4 + x^8)^(1/4)/(Sqrt
[-Sqrt[1 - I*Sqrt[31]]] + Sqrt[2]*x), x])/Sqrt[-31*Sqrt[1 - I*Sqrt[31]]] - ((I + Sqrt[31])*Defer[Int][(2 - 2*x
^4 + x^8)^(1/4)/(Sqrt[-Sqrt[1 - I*Sqrt[31]]] + Sqrt[2]*x), x])/(2*Sqrt[-31*Sqrt[1 - I*Sqrt[31]]]) + ((I/2)*Def
er[Int][(2 - 2*x^4 + x^8)^(1/4)/((1 - I*Sqrt[31])^(1/4) + Sqrt[2]*x), x])/(Sqrt[31]*(1 - I*Sqrt[31])^(1/4)) -
((I/2)*(1 - I*Sqrt[31])^(3/4)*Defer[Int][(2 - 2*x^4 + x^8)^(1/4)/((1 - I*Sqrt[31])^(1/4) + Sqrt[2]*x), x])/Sqr
t[31] - ((I/2)*Defer[Int][(2 - 2*x^4 + x^8)^(1/4)/(Sqrt[-Sqrt[1 + I*Sqrt[31]]] + Sqrt[2]*x), x])/Sqrt[-31*Sqrt
[1 + I*Sqrt[31]]] + ((I - Sqrt[31])*Defer[Int][(2 - 2*x^4 + x^8)^(1/4)/(Sqrt[-Sqrt[1 + I*Sqrt[31]]] + Sqrt[2]*
x), x])/(2*Sqrt[-31*Sqrt[1 + I*Sqrt[31]]]) - ((I/2)*Defer[Int][(2 - 2*x^4 + x^8)^(1/4)/((1 + I*Sqrt[31])^(1/4)
+ Sqrt[2]*x), x])/(Sqrt[31]*(1 + I*Sqrt[31])^(1/4)) + ((I/2)*(1 + I*Sqrt[31])^(3/4)*Defer[Int][(2 - 2*x^4 + x
^8)^(1/4)/((1 + I*Sqrt[31])^(1/4) + Sqrt[2]*x), x])/Sqrt[31]

Rubi steps

\begin {align*} \int \frac {x^2 \left (-2+x^8\right ) \sqrt [4]{2-2 x^4+x^8}}{\left (2+x^8\right ) \left (4-x^4+2 x^8\right )} \, dx &=\int \left (-\frac {2 x^6 \sqrt [4]{2-2 x^4+x^8}}{2+x^8}+\frac {x^2 \left (-1+4 x^4\right ) \sqrt [4]{2-2 x^4+x^8}}{4-x^4+2 x^8}\right ) \, dx\\ &=-\left (2 \int \frac {x^6 \sqrt [4]{2-2 x^4+x^8}}{2+x^8} \, dx\right )+\int \frac {x^2 \left (-1+4 x^4\right ) \sqrt [4]{2-2 x^4+x^8}}{4-x^4+2 x^8} \, dx\\ &=-\left (2 \int \left (\frac {x^2 \sqrt [4]{2-2 x^4+x^8}}{2 \left (-i \sqrt {2}+x^4\right )}+\frac {x^2 \sqrt [4]{2-2 x^4+x^8}}{2 \left (i \sqrt {2}+x^4\right )}\right ) \, dx\right )+\int \left (-\frac {x^2 \sqrt [4]{2-2 x^4+x^8}}{4-x^4+2 x^8}+\frac {4 x^6 \sqrt [4]{2-2 x^4+x^8}}{4-x^4+2 x^8}\right ) \, dx\\ &=4 \int \frac {x^6 \sqrt [4]{2-2 x^4+x^8}}{4-x^4+2 x^8} \, dx-\int \frac {x^2 \sqrt [4]{2-2 x^4+x^8}}{-i \sqrt {2}+x^4} \, dx-\int \frac {x^2 \sqrt [4]{2-2 x^4+x^8}}{i \sqrt {2}+x^4} \, dx-\int \frac {x^2 \sqrt [4]{2-2 x^4+x^8}}{4-x^4+2 x^8} \, dx\\ &=4 \int \left (\frac {i \left (1+i \sqrt {31}\right ) x^2 \sqrt [4]{2-2 x^4+x^8}}{\sqrt {31} \left (1+i \sqrt {31}-4 x^4\right )}-\frac {i \left (-1+i \sqrt {31}\right ) x^2 \sqrt [4]{2-2 x^4+x^8}}{\sqrt {31} \left (-1+i \sqrt {31}+4 x^4\right )}\right ) \, dx-\int \left (-\frac {\sqrt [4]{2-2 x^4+x^8}}{2 \left (\sqrt [4]{-2}-x^2\right )}+\frac {\sqrt [4]{2-2 x^4+x^8}}{2 \left (\sqrt [4]{-2}+x^2\right )}\right ) \, dx-\int \left (-\frac {\sqrt [4]{2-2 x^4+x^8}}{2 \left (-(-1)^{3/4} \sqrt [4]{2}-x^2\right )}+\frac {\sqrt [4]{2-2 x^4+x^8}}{2 \left (-(-1)^{3/4} \sqrt [4]{2}+x^2\right )}\right ) \, dx-\int \left (\frac {4 i x^2 \sqrt [4]{2-2 x^4+x^8}}{\sqrt {31} \left (1+i \sqrt {31}-4 x^4\right )}+\frac {4 i x^2 \sqrt [4]{2-2 x^4+x^8}}{\sqrt {31} \left (-1+i \sqrt {31}+4 x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [4]{-2}-x^2} \, dx+\frac {1}{2} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{-(-1)^{3/4} \sqrt [4]{2}-x^2} \, dx-\frac {1}{2} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [4]{-2}+x^2} \, dx-\frac {1}{2} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{-(-1)^{3/4} \sqrt [4]{2}+x^2} \, dx-\frac {(4 i) \int \frac {x^2 \sqrt [4]{2-2 x^4+x^8}}{1+i \sqrt {31}-4 x^4} \, dx}{\sqrt {31}}-\frac {(4 i) \int \frac {x^2 \sqrt [4]{2-2 x^4+x^8}}{-1+i \sqrt {31}+4 x^4} \, dx}{\sqrt {31}}-\frac {1}{31} \left (4 \left (31-i \sqrt {31}\right )\right ) \int \frac {x^2 \sqrt [4]{2-2 x^4+x^8}}{1+i \sqrt {31}-4 x^4} \, dx+\frac {1}{31} \left (4 \left (31+i \sqrt {31}\right )\right ) \int \frac {x^2 \sqrt [4]{2-2 x^4+x^8}}{-1+i \sqrt {31}+4 x^4} \, dx\\ &=\frac {1}{2} \int \left (-\frac {(-1)^{7/8} \sqrt [4]{2-2 x^4+x^8}}{2 \sqrt [8]{2} \left (\sqrt [8]{-2}-x\right )}-\frac {(-1)^{7/8} \sqrt [4]{2-2 x^4+x^8}}{2 \sqrt [8]{2} \left (\sqrt [8]{-2}+x\right )}\right ) \, dx-\frac {1}{2} \int \left (-\frac {(-1)^{3/8} \sqrt [4]{2-2 x^4+x^8}}{2 \sqrt [8]{2} \left (-(-1)^{5/8} \sqrt [8]{2}-x\right )}-\frac {(-1)^{3/8} \sqrt [4]{2-2 x^4+x^8}}{2 \sqrt [8]{2} \left (-(-1)^{5/8} \sqrt [8]{2}+x\right )}\right ) \, dx-\frac {1}{2} \int \left (\frac {\sqrt {-1+i} \sqrt [4]{-\frac {1}{2}} \sqrt [4]{2-2 x^4+x^8}}{2 \left (\sqrt {-1+i}-\sqrt [8]{2} x\right )}+\frac {\sqrt {-1+i} \sqrt [4]{-\frac {1}{2}} \sqrt [4]{2-2 x^4+x^8}}{2 \left (\sqrt {-1+i}+\sqrt [8]{2} x\right )}\right ) \, dx+\frac {1}{2} \int \left (\frac {\sqrt [4]{-\frac {1}{2}} \sqrt {1-i} \sqrt [4]{2-2 x^4+x^8}}{2 \left (\sqrt {1-i}-\sqrt [8]{2} x\right )}+\frac {\sqrt [4]{-\frac {1}{2}} \sqrt {1-i} \sqrt [4]{2-2 x^4+x^8}}{2 \left (\sqrt {1-i}+\sqrt [8]{2} x\right )}\right ) \, dx-\frac {(4 i) \int \left (-\frac {\sqrt [4]{2-2 x^4+x^8}}{4 \left (\sqrt {1-i \sqrt {31}}-2 x^2\right )}+\frac {\sqrt [4]{2-2 x^4+x^8}}{4 \left (\sqrt {1-i \sqrt {31}}+2 x^2\right )}\right ) \, dx}{\sqrt {31}}-\frac {(4 i) \int \left (\frac {\sqrt [4]{2-2 x^4+x^8}}{4 \left (\sqrt {1+i \sqrt {31}}-2 x^2\right )}-\frac {\sqrt [4]{2-2 x^4+x^8}}{4 \left (\sqrt {1+i \sqrt {31}}+2 x^2\right )}\right ) \, dx}{\sqrt {31}}-\frac {1}{31} \left (4 \left (31-i \sqrt {31}\right )\right ) \int \left (\frac {\sqrt [4]{2-2 x^4+x^8}}{4 \left (\sqrt {1+i \sqrt {31}}-2 x^2\right )}-\frac {\sqrt [4]{2-2 x^4+x^8}}{4 \left (\sqrt {1+i \sqrt {31}}+2 x^2\right )}\right ) \, dx+\frac {1}{31} \left (4 \left (31+i \sqrt {31}\right )\right ) \int \left (-\frac {\sqrt [4]{2-2 x^4+x^8}}{4 \left (\sqrt {1-i \sqrt {31}}-2 x^2\right )}+\frac {\sqrt [4]{2-2 x^4+x^8}}{4 \left (\sqrt {1-i \sqrt {31}}+2 x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {-1+i}-\sqrt [8]{2} x} \, dx}{2 \sqrt {-1+i} 2^{3/4}}+\frac {\int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {-1+i}+\sqrt [8]{2} x} \, dx}{2 \sqrt {-1+i} 2^{3/4}}+\frac {\int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {1-i}-\sqrt [8]{2} x} \, dx}{2 \sqrt {1-i} 2^{3/4}}+\frac {\int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {1-i}+\sqrt [8]{2} x} \, dx}{2 \sqrt {1-i} 2^{3/4}}+\frac {(-1)^{3/8} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{-(-1)^{5/8} \sqrt [8]{2}-x} \, dx}{4 \sqrt [8]{2}}+\frac {(-1)^{3/8} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{-(-1)^{5/8} \sqrt [8]{2}+x} \, dx}{4 \sqrt [8]{2}}-\frac {(-1)^{7/8} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [8]{-2}-x} \, dx}{4 \sqrt [8]{2}}-\frac {(-1)^{7/8} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [8]{-2}+x} \, dx}{4 \sqrt [8]{2}}+\frac {i \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {1-i \sqrt {31}}-2 x^2} \, dx}{\sqrt {31}}-\frac {i \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {1+i \sqrt {31}}-2 x^2} \, dx}{\sqrt {31}}-\frac {i \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {1-i \sqrt {31}}+2 x^2} \, dx}{\sqrt {31}}+\frac {i \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {1+i \sqrt {31}}+2 x^2} \, dx}{\sqrt {31}}-\frac {1}{31} \left (31-i \sqrt {31}\right ) \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {1+i \sqrt {31}}-2 x^2} \, dx-\frac {1}{31} \left (-31+i \sqrt {31}\right ) \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {1+i \sqrt {31}}+2 x^2} \, dx-\frac {1}{31} \left (31+i \sqrt {31}\right ) \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {1-i \sqrt {31}}-2 x^2} \, dx+\frac {1}{31} \left (31+i \sqrt {31}\right ) \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {1-i \sqrt {31}}+2 x^2} \, dx\\ &=\frac {\int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {-1+i}-\sqrt [8]{2} x} \, dx}{2 \sqrt {-1+i} 2^{3/4}}+\frac {\int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {-1+i}+\sqrt [8]{2} x} \, dx}{2 \sqrt {-1+i} 2^{3/4}}+\frac {\int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {1-i}-\sqrt [8]{2} x} \, dx}{2 \sqrt {1-i} 2^{3/4}}+\frac {\int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {1-i}+\sqrt [8]{2} x} \, dx}{2 \sqrt {1-i} 2^{3/4}}+\frac {(-1)^{3/8} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{-(-1)^{5/8} \sqrt [8]{2}-x} \, dx}{4 \sqrt [8]{2}}+\frac {(-1)^{3/8} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{-(-1)^{5/8} \sqrt [8]{2}+x} \, dx}{4 \sqrt [8]{2}}-\frac {(-1)^{7/8} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [8]{-2}-x} \, dx}{4 \sqrt [8]{2}}-\frac {(-1)^{7/8} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [8]{-2}+x} \, dx}{4 \sqrt [8]{2}}-\frac {i \int \left (\frac {\sqrt {-\sqrt {1-i \sqrt {31}}} \sqrt [4]{2-2 x^4+x^8}}{2 \sqrt {1-i \sqrt {31}} \left (\sqrt {-\sqrt {1-i \sqrt {31}}}-\sqrt {2} x\right )}+\frac {\sqrt {-\sqrt {1-i \sqrt {31}}} \sqrt [4]{2-2 x^4+x^8}}{2 \sqrt {1-i \sqrt {31}} \left (\sqrt {-\sqrt {1-i \sqrt {31}}}+\sqrt {2} x\right )}\right ) \, dx}{\sqrt {31}}+\frac {i \int \left (\frac {\sqrt [4]{2-2 x^4+x^8}}{2 \sqrt [4]{1-i \sqrt {31}} \left (\sqrt [4]{1-i \sqrt {31}}-\sqrt {2} x\right )}+\frac {\sqrt [4]{2-2 x^4+x^8}}{2 \sqrt [4]{1-i \sqrt {31}} \left (\sqrt [4]{1-i \sqrt {31}}+\sqrt {2} x\right )}\right ) \, dx}{\sqrt {31}}+\frac {i \int \left (\frac {\sqrt {-\sqrt {1+i \sqrt {31}}} \sqrt [4]{2-2 x^4+x^8}}{2 \sqrt {1+i \sqrt {31}} \left (\sqrt {-\sqrt {1+i \sqrt {31}}}-\sqrt {2} x\right )}+\frac {\sqrt {-\sqrt {1+i \sqrt {31}}} \sqrt [4]{2-2 x^4+x^8}}{2 \sqrt {1+i \sqrt {31}} \left (\sqrt {-\sqrt {1+i \sqrt {31}}}+\sqrt {2} x\right )}\right ) \, dx}{\sqrt {31}}-\frac {i \int \left (\frac {\sqrt [4]{2-2 x^4+x^8}}{2 \sqrt [4]{1+i \sqrt {31}} \left (\sqrt [4]{1+i \sqrt {31}}-\sqrt {2} x\right )}+\frac {\sqrt [4]{2-2 x^4+x^8}}{2 \sqrt [4]{1+i \sqrt {31}} \left (\sqrt [4]{1+i \sqrt {31}}+\sqrt {2} x\right )}\right ) \, dx}{\sqrt {31}}-\frac {1}{31} \left (31-i \sqrt {31}\right ) \int \left (\frac {\sqrt [4]{2-2 x^4+x^8}}{2 \sqrt [4]{1+i \sqrt {31}} \left (\sqrt [4]{1+i \sqrt {31}}-\sqrt {2} x\right )}+\frac {\sqrt [4]{2-2 x^4+x^8}}{2 \sqrt [4]{1+i \sqrt {31}} \left (\sqrt [4]{1+i \sqrt {31}}+\sqrt {2} x\right )}\right ) \, dx-\frac {1}{31} \left (-31+i \sqrt {31}\right ) \int \left (\frac {\sqrt {-\sqrt {1+i \sqrt {31}}} \sqrt [4]{2-2 x^4+x^8}}{2 \sqrt {1+i \sqrt {31}} \left (\sqrt {-\sqrt {1+i \sqrt {31}}}-\sqrt {2} x\right )}+\frac {\sqrt {-\sqrt {1+i \sqrt {31}}} \sqrt [4]{2-2 x^4+x^8}}{2 \sqrt {1+i \sqrt {31}} \left (\sqrt {-\sqrt {1+i \sqrt {31}}}+\sqrt {2} x\right )}\right ) \, dx+\frac {1}{31} \left (31+i \sqrt {31}\right ) \int \left (\frac {\sqrt {-\sqrt {1-i \sqrt {31}}} \sqrt [4]{2-2 x^4+x^8}}{2 \sqrt {1-i \sqrt {31}} \left (\sqrt {-\sqrt {1-i \sqrt {31}}}-\sqrt {2} x\right )}+\frac {\sqrt {-\sqrt {1-i \sqrt {31}}} \sqrt [4]{2-2 x^4+x^8}}{2 \sqrt {1-i \sqrt {31}} \left (\sqrt {-\sqrt {1-i \sqrt {31}}}+\sqrt {2} x\right )}\right ) \, dx-\frac {1}{31} \left (31+i \sqrt {31}\right ) \int \left (\frac {\sqrt [4]{2-2 x^4+x^8}}{2 \sqrt [4]{1-i \sqrt {31}} \left (\sqrt [4]{1-i \sqrt {31}}-\sqrt {2} x\right )}+\frac {\sqrt [4]{2-2 x^4+x^8}}{2 \sqrt [4]{1-i \sqrt {31}} \left (\sqrt [4]{1-i \sqrt {31}}+\sqrt {2} x\right )}\right ) \, dx\\ &=\frac {\int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {-1+i}-\sqrt [8]{2} x} \, dx}{2 \sqrt {-1+i} 2^{3/4}}+\frac {\int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {-1+i}+\sqrt [8]{2} x} \, dx}{2 \sqrt {-1+i} 2^{3/4}}+\frac {\int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {1-i}-\sqrt [8]{2} x} \, dx}{2 \sqrt {1-i} 2^{3/4}}+\frac {\int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {1-i}+\sqrt [8]{2} x} \, dx}{2 \sqrt {1-i} 2^{3/4}}+\frac {(-1)^{3/8} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{-(-1)^{5/8} \sqrt [8]{2}-x} \, dx}{4 \sqrt [8]{2}}+\frac {(-1)^{3/8} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{-(-1)^{5/8} \sqrt [8]{2}+x} \, dx}{4 \sqrt [8]{2}}-\frac {(-1)^{7/8} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [8]{-2}-x} \, dx}{4 \sqrt [8]{2}}-\frac {(-1)^{7/8} \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [8]{-2}+x} \, dx}{4 \sqrt [8]{2}}+\frac {i \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {-\sqrt {1-i \sqrt {31}}}-\sqrt {2} x} \, dx}{2 \sqrt {-31 \sqrt {1-i \sqrt {31}}}}+\frac {i \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {-\sqrt {1-i \sqrt {31}}}+\sqrt {2} x} \, dx}{2 \sqrt {-31 \sqrt {1-i \sqrt {31}}}}+\frac {i \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [4]{1-i \sqrt {31}}-\sqrt {2} x} \, dx}{2 \sqrt {31} \sqrt [4]{1-i \sqrt {31}}}+\frac {i \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [4]{1-i \sqrt {31}}+\sqrt {2} x} \, dx}{2 \sqrt {31} \sqrt [4]{1-i \sqrt {31}}}-\frac {\left (i \left (1-i \sqrt {31}\right )^{3/4}\right ) \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [4]{1-i \sqrt {31}}-\sqrt {2} x} \, dx}{2 \sqrt {31}}-\frac {\left (i \left (1-i \sqrt {31}\right )^{3/4}\right ) \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [4]{1-i \sqrt {31}}+\sqrt {2} x} \, dx}{2 \sqrt {31}}-\frac {i \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {-\sqrt {1+i \sqrt {31}}}-\sqrt {2} x} \, dx}{2 \sqrt {-31 \sqrt {1+i \sqrt {31}}}}-\frac {i \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {-\sqrt {1+i \sqrt {31}}}+\sqrt {2} x} \, dx}{2 \sqrt {-31 \sqrt {1+i \sqrt {31}}}}+\frac {\left (i-\sqrt {31}\right ) \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {-\sqrt {1+i \sqrt {31}}}-\sqrt {2} x} \, dx}{2 \sqrt {-31 \sqrt {1+i \sqrt {31}}}}+\frac {\left (i-\sqrt {31}\right ) \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {-\sqrt {1+i \sqrt {31}}}+\sqrt {2} x} \, dx}{2 \sqrt {-31 \sqrt {1+i \sqrt {31}}}}-\frac {i \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [4]{1+i \sqrt {31}}-\sqrt {2} x} \, dx}{2 \sqrt {31} \sqrt [4]{1+i \sqrt {31}}}-\frac {i \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [4]{1+i \sqrt {31}}+\sqrt {2} x} \, dx}{2 \sqrt {31} \sqrt [4]{1+i \sqrt {31}}}+\frac {\left (i \left (1+i \sqrt {31}\right )^{3/4}\right ) \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [4]{1+i \sqrt {31}}-\sqrt {2} x} \, dx}{2 \sqrt {31}}+\frac {\left (i \left (1+i \sqrt {31}\right )^{3/4}\right ) \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt [4]{1+i \sqrt {31}}+\sqrt {2} x} \, dx}{2 \sqrt {31}}-\frac {\left (i+\sqrt {31}\right ) \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {-\sqrt {1-i \sqrt {31}}}-\sqrt {2} x} \, dx}{2 \sqrt {-31 \sqrt {1-i \sqrt {31}}}}-\frac {\left (i+\sqrt {31}\right ) \int \frac {\sqrt [4]{2-2 x^4+x^8}}{\sqrt {-\sqrt {1-i \sqrt {31}}}+\sqrt {2} x} \, dx}{2 \sqrt {-31 \sqrt {1-i \sqrt {31}}}}\\ \end {align*}

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Mathematica [F]  time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (-2+x^8\right ) \sqrt [4]{2-2 x^4+x^8}}{\left (2+x^8\right ) \left (4-x^4+2 x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^2*(-2 + x^8)*(2 - 2*x^4 + x^8)^(1/4))/((2 + x^8)*(4 - x^4 + 2*x^8)),x]

[Out]

Integrate[(x^2*(-2 + x^8)*(2 - 2*x^4 + x^8)^(1/4))/((2 + x^8)*(4 - x^4 + 2*x^8)), x]

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IntegrateAlgebraic [A]  time = 1.78, size = 253, normalized size = 1.02 \begin {gather*} \frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{2-2 x^4+x^8}}{\sqrt {2} x^2-\sqrt {2-2 x^4+x^8}}\right )}{2 \sqrt [4]{2}}-\frac {\sqrt [4]{3} \tan ^{-1}\left (\frac {-\frac {\sqrt [4]{3} x^2}{2^{3/4}}+\frac {\sqrt {2-2 x^4+x^8}}{\sqrt [4]{6}}}{x \sqrt [4]{2-2 x^4+x^8}}\right )}{2\ 2^{3/4}}+\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{2-2 x^4+x^8}}{2 x^2+\sqrt {2} \sqrt {2-2 x^4+x^8}}\right )}{2 \sqrt [4]{2}}-\frac {\sqrt [4]{3} \tanh ^{-1}\left (\frac {6^{3/4} x \sqrt [4]{2-2 x^4+x^8}}{3 x^2+\sqrt {6} \sqrt {2-2 x^4+x^8}}\right )}{2\ 2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^2*(-2 + x^8)*(2 - 2*x^4 + x^8)^(1/4))/((2 + x^8)*(4 - x^4 + 2*x^8)),x]

[Out]

ArcTan[(2^(3/4)*x*(2 - 2*x^4 + x^8)^(1/4))/(Sqrt[2]*x^2 - Sqrt[2 - 2*x^4 + x^8])]/(2*2^(1/4)) - (3^(1/4)*ArcTa
n[(-((3^(1/4)*x^2)/2^(3/4)) + Sqrt[2 - 2*x^4 + x^8]/6^(1/4))/(x*(2 - 2*x^4 + x^8)^(1/4))])/(2*2^(3/4)) + ArcTa
nh[(2*2^(1/4)*x*(2 - 2*x^4 + x^8)^(1/4))/(2*x^2 + Sqrt[2]*Sqrt[2 - 2*x^4 + x^8])]/(2*2^(1/4)) - (3^(1/4)*ArcTa
nh[(6^(3/4)*x*(2 - 2*x^4 + x^8)^(1/4))/(3*x^2 + Sqrt[6]*Sqrt[2 - 2*x^4 + x^8])])/(2*2^(3/4))

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fricas [B]  time = 24.39, size = 1328, normalized size = 5.35

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^8-2)*(x^8-2*x^4+2)^(1/4)/(x^8+2)/(2*x^8-x^4+4),x, algorithm="fricas")

[Out]

-1/16*8^(3/4)*sqrt(2)*arctan(1/4*(4*8^(1/4)*sqrt(2)*(x^8 - 2*x^4 + 2)^(1/4)*x^3 + 8^(3/4)*sqrt(2)*(x^8 - 2*x^4
+ 2)^(3/4)*x + (8^(3/4)*sqrt(2)*(x^8 - 2*x^4 + 2)^(1/4)*x^3 + 2*8^(1/4)*sqrt(2)*(x^8 - 2*x^4 + 2)^(3/4)*x - 8
*sqrt(x^8 - 2*x^4 + 2)*x^2 - 2*sqrt(2)*(x^8 + 2))*sqrt((2*x^8 + 4*8^(1/4)*sqrt(2)*(x^8 - 2*x^4 + 2)^(1/4)*x^3
+ 8^(3/4)*sqrt(2)*(x^8 - 2*x^4 + 2)^(3/4)*x + 8*sqrt(2)*sqrt(x^8 - 2*x^4 + 2)*x^2 + 4)/(x^8 + 2)))/(x^8 - 4*x^
4 + 2)) - 1/16*8^(3/4)*sqrt(2)*arctan(1/4*(4*8^(1/4)*sqrt(2)*(x^8 - 2*x^4 + 2)^(1/4)*x^3 + 8^(3/4)*sqrt(2)*(x^
8 - 2*x^4 + 2)^(3/4)*x + (8^(3/4)*sqrt(2)*(x^8 - 2*x^4 + 2)^(1/4)*x^3 + 2*8^(1/4)*sqrt(2)*(x^8 - 2*x^4 + 2)^(3
/4)*x + 8*sqrt(x^8 - 2*x^4 + 2)*x^2 + 2*sqrt(2)*(x^8 + 2))*sqrt((2*x^8 - 4*8^(1/4)*sqrt(2)*(x^8 - 2*x^4 + 2)^(
1/4)*x^3 - 8^(3/4)*sqrt(2)*(x^8 - 2*x^4 + 2)^(3/4)*x + 8*sqrt(2)*sqrt(x^8 - 2*x^4 + 2)*x^2 + 4)/(x^8 + 2)))/(x
^8 - 4*x^4 + 2)) + 1/64*8^(3/4)*sqrt(2)*log(4*(2*x^8 + 4*8^(1/4)*sqrt(2)*(x^8 - 2*x^4 + 2)^(1/4)*x^3 + 8^(3/4)
*sqrt(2)*(x^8 - 2*x^4 + 2)^(3/4)*x + 8*sqrt(2)*sqrt(x^8 - 2*x^4 + 2)*x^2 + 4)/(x^8 + 2)) - 1/64*8^(3/4)*sqrt(2
)*log(4*(2*x^8 - 4*8^(1/4)*sqrt(2)*(x^8 - 2*x^4 + 2)^(1/4)*x^3 - 8^(3/4)*sqrt(2)*(x^8 - 2*x^4 + 2)^(3/4)*x + 8
*sqrt(2)*sqrt(x^8 - 2*x^4 + 2)*x^2 + 4)/(x^8 + 2)) + 1/4*3^(1/4)*2^(1/4)*arctan(1/6*(6*3^(3/4)*2^(3/4)*(x^8 -
2*x^4 + 2)^(1/4)*x^3 + 12*3^(1/4)*2^(1/4)*(x^8 - 2*x^4 + 2)^(3/4)*x + sqrt(6)*(6*3^(1/4)*2^(1/4)*(x^8 - 2*x^4
+ 2)^(1/4)*x^3 + 2*3^(3/4)*2^(3/4)*(x^8 - 2*x^4 + 2)^(3/4)*x - 12*sqrt(x^8 - 2*x^4 + 2)*x^2 - sqrt(3)*sqrt(2)*
(2*x^8 - x^4 + 4))*sqrt((2*x^8 + 2*3^(3/4)*2^(3/4)*(x^8 - 2*x^4 + 2)^(1/4)*x^3 - x^4 + 4*sqrt(3)*sqrt(2)*sqrt(
x^8 - 2*x^4 + 2)*x^2 + 4*3^(1/4)*2^(1/4)*(x^8 - 2*x^4 + 2)^(3/4)*x + 4)/(2*x^8 - x^4 + 4)))/(2*x^8 - 7*x^4 + 4
)) + 1/4*3^(1/4)*2^(1/4)*arctan(1/6*(6*3^(3/4)*2^(3/4)*(x^8 - 2*x^4 + 2)^(1/4)*x^3 + 12*3^(1/4)*2^(1/4)*(x^8 -
2*x^4 + 2)^(3/4)*x + sqrt(6)*(6*3^(1/4)*2^(1/4)*(x^8 - 2*x^4 + 2)^(1/4)*x^3 + 2*3^(3/4)*2^(3/4)*(x^8 - 2*x^4
+ 2)^(3/4)*x + 12*sqrt(x^8 - 2*x^4 + 2)*x^2 + sqrt(3)*sqrt(2)*(2*x^8 - x^4 + 4))*sqrt((2*x^8 - 2*3^(3/4)*2^(3/
4)*(x^8 - 2*x^4 + 2)^(1/4)*x^3 - x^4 + 4*sqrt(3)*sqrt(2)*sqrt(x^8 - 2*x^4 + 2)*x^2 - 4*3^(1/4)*2^(1/4)*(x^8 -
2*x^4 + 2)^(3/4)*x + 4)/(2*x^8 - x^4 + 4)))/(2*x^8 - 7*x^4 + 4)) - 1/16*3^(1/4)*2^(1/4)*log(6*(2*x^8 + 2*3^(3/
4)*2^(3/4)*(x^8 - 2*x^4 + 2)^(1/4)*x^3 - x^4 + 4*sqrt(3)*sqrt(2)*sqrt(x^8 - 2*x^4 + 2)*x^2 + 4*3^(1/4)*2^(1/4)
*(x^8 - 2*x^4 + 2)^(3/4)*x + 4)/(2*x^8 - x^4 + 4)) + 1/16*3^(1/4)*2^(1/4)*log(6*(2*x^8 - 2*3^(3/4)*2^(3/4)*(x^
8 - 2*x^4 + 2)^(1/4)*x^3 - x^4 + 4*sqrt(3)*sqrt(2)*sqrt(x^8 - 2*x^4 + 2)*x^2 - 4*3^(1/4)*2^(1/4)*(x^8 - 2*x^4
+ 2)^(3/4)*x + 4)/(2*x^8 - x^4 + 4))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} - 2 \, x^{4} + 2\right )}^{\frac {1}{4}} {\left (x^{8} - 2\right )} x^{2}}{{\left (2 \, x^{8} - x^{4} + 4\right )} {\left (x^{8} + 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^8-2)*(x^8-2*x^4+2)^(1/4)/(x^8+2)/(2*x^8-x^4+4),x, algorithm="giac")

[Out]

integrate((x^8 - 2*x^4 + 2)^(1/4)*(x^8 - 2)*x^2/((2*x^8 - x^4 + 4)*(x^8 + 2)), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (x^{8}-2\right ) \left (x^{8}-2 x^{4}+2\right )^{\frac {1}{4}}}{\left (x^{8}+2\right ) \left (2 x^{8}-x^{4}+4\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^8-2)*(x^8-2*x^4+2)^(1/4)/(x^8+2)/(2*x^8-x^4+4),x)

[Out]

int(x^2*(x^8-2)*(x^8-2*x^4+2)^(1/4)/(x^8+2)/(2*x^8-x^4+4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} - 2 \, x^{4} + 2\right )}^{\frac {1}{4}} {\left (x^{8} - 2\right )} x^{2}}{{\left (2 \, x^{8} - x^{4} + 4\right )} {\left (x^{8} + 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^8-2)*(x^8-2*x^4+2)^(1/4)/(x^8+2)/(2*x^8-x^4+4),x, algorithm="maxima")

[Out]

integrate((x^8 - 2*x^4 + 2)^(1/4)*(x^8 - 2)*x^2/((2*x^8 - x^4 + 4)*(x^8 + 2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,\left (x^8-2\right )\,{\left (x^8-2\,x^4+2\right )}^{1/4}}{\left (x^8+2\right )\,\left (2\,x^8-x^4+4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(x^8 - 2)*(x^8 - 2*x^4 + 2)^(1/4))/((x^8 + 2)*(2*x^8 - x^4 + 4)),x)

[Out]

int((x^2*(x^8 - 2)*(x^8 - 2*x^4 + 2)^(1/4))/((x^8 + 2)*(2*x^8 - x^4 + 4)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(x**8-2)*(x**8-2*x**4+2)**(1/4)/(x**8+2)/(2*x**8-x**4+4),x)

[Out]

Timed out

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